Finite Group Databasebeta

← #5956450

Subgroups — #5956450

Order 512.

3404 conjugacy classes of subgroups.

The subgroup lattice has 689 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2327C211yesmaximal, characteristic, complemented
2C2×#2323C212yesmaximal, complemented
3C2×C4×C22⋊Q8C212yesmaximal, complemented
4C2×#2290C211yesmaximal, characteristic
5#15223C211yesmaximal, characteristic, complemented
6#15223C211yesmaximal, characteristic, complemented
7#15193C212yesmaximal, complemented
8#15178C211yesmaximal, characteristic, complemented
9#15178C211yesmaximal, characteristic, complemented
10#15177C212yesmaximal
11#15082C214yesmaximal, complemented
12#15079C214yesmaximal, complemented
13#14985C214yesmaximal
14#14969C214yesmaximal, complemented
15#13797C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C22×C22⋊Q8C2212yes—
19C22×C4×Q8C2212yes—
20C22×C4×D4C2211yescharacteristic
21C22×#117C2211yescharacteristic
22C23×C42C2211yescharacteristic
23#2327—28no—
24#2323—216no—
25C4×C22⋊Q8—216no—
26C2×#391C2212yes—
27C2×#391C2212yes—
28C2×#391C2212yes—
29C2×#389C2212yes—
30C2×#389C2214yes—
31C2×#389C2212yes—
32#2290—28no—
33C2×#388C2214yes—
34C2×#388C2214yes—
35C2×#388C2212yes—
36C2×#387C2214yes—
37C2×#387C2214yes—
38C2×#387C2214yes—
39C2×#387C2212yes—
40C2×#387C2214yes—
41C2×#386C2212yes—
42C2×#385C2212yes—
43C2×#385C2214yes—
44C2×#382C2211yescharacteristic
45C2×#380C2212yes—
46C2×#380C2212yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2214yes—
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C22⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4C2214yes—
53C2×#376C2211yescharacteristic
54#1445C2214yes—
55#1445C2214yes—
56#1445C2214yes—
57#1445C2214yes—
58#1445C2214yes—
59#1444C2214yes—
60#1444C2214yes—
61#1444C2214yes—
62#1444C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1438C2214yes—
70C4⋊#95C2212yes—
71C4⋊#95C2211yescharacteristic
72C4⋊#95C2211yescharacteristic
73#1434C2214yes—
74#1434C2214yes—
75C23×Q8C2312yes—
76C23×D4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2312yes—
80C2×C4⋊Q8—28no—
81C2×C4⋊Q8—24no—
82C2×C22⋊Q8—28no—
83C2×C22⋊Q8—28no—
84C2×C22⋊Q8—28no—
85C2×C22⋊Q8—28no—
86C2×C22⋊Q8—28no—
87C2×C22⋊Q8—28no—
88C2×C4×Q8—28no—
89C2×C4×Q8—28no—
90C2×C4×Q8—28no—
91C2×C4×D4—24no—
92C2×C4×D4—24no—
93C2×C4×D4—24no—
94C2×#117—28no—
95C2×#117—24no—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2311yescharacteristic
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C42—24no—
118C22×C42—28no—
119C22×C42C2311yescharacteristic
120C22×C42C2312yes—
121C22×C42C2311yescharacteristic
122C22×C42C2314yes—
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2314yes—
126C22×C42C2311yescharacteristic
127#391—28no—
128#391—28no—
129#391—28no—
130#389—28no—
131#389—28no—
132#389—216no—
133#388—216no—
134#388—216no—
135#388—28no—
136#387—216no—
137#387—216no—
138#387—216no—
139#387—28no—
140#387—216no—
141#386—28no—
142#385—28no—
143#385—216no—
144#382—24no—
145#380—28no—
146#380—28no—
147C4×C4⋊C4—24no—
148C4×C4⋊C4—216no—
149C4×C4⋊C4D4116yescomplemented
150C4×C22⋊C4—24no—
151C4×C22⋊C4—24no—
152C4×C22⋊C4—216no—
153#376D418yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C25C2411yescharacteristic
176C25C2411yescharacteristic
177C22×Q8—24no—
178C22×Q8—24no—
179C22×Q8—24no—
180C22×Q8—24no—
181C22×Q8—24no—
182C22×Q8—24no—
183C22×Q8#4118yes—
184C22×D4—22no—
185C22×D4—22no—
186C22×D4—24no—
187C22×D4—22no—
188C22×D4—42no—
189C23×C4—24no—
190C23×C4—22no—
191C23×C4—24no—
192C23×C4—22no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4#4114yes—
205C23×C4#4114yes—
206C23×C4C2412yes—
207C23×C4C2414yes—
208C23×C4C2414yes—
209C23×C4C2414yes—
210C23×C4C2411yescharacteristic
211C23×C4C2412yes—
212C23×C4C2411yescharacteristic
213C23×C4C2414yes—
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C4⋊Q8—232no—
220C22⋊Q8—264no—
221C22⋊Q8—264no—
222C4×Q8—432no—
223C4×D4—232no—
224#117—416no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4C2×D418yes—
260C2×C4⋊C4C2×D418yes—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—24no—
291C2×C42—22no—
292C2×C42—24no—
293C2×C42—28no—
294C2×C42—28no—
295C2×C42—28no—
296C2×C42—22no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—44no—
302C2×C42—44no—
303C2×C42—416no—
304C2×C42C2×D414yes—
305C2×C42C2×D414yes—
306C2×C42C2×D4116yes—
307C2×C42C2×D414yes—
308C2×C42C2×D418yes—
309#95—24no—
310#95—24no—
311#95—28no—
312#95—28no—
313#95—28no—
314#95—28no—
315#95—24no—
316#95—28no—
317#95—28no—
318#95—28no—
319#95—28no—
320#95—28no—
321#95—28no—
322#95—28no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95C2×D418yes—
328#95C2×D418yes—
329#95C2×D4116yes—
330C24—22no—
331C24—21no—
332C24—22no—
333C24—21no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—21no—
339C24—22no—
340C24—21no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24C2×#4112yes—
346C24C2×#4112yes—
347C242−1+414yes—
348C24C2511yescharacteristic
349C2×Q8—216no—
350C2×Q8—216no—
351C2×Q8—216no—
352C2×Q8—48no—
353C2×Q8—48no—
354C2×Q8—48no—
355C2×Q8—48no—
356C2×D4—28no—
357C2×D4—48no—
358C2×D4—48no—
359C2×D4—44no—
360C2×D4—44no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—216no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—216no—
415C22×C4—216no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D414yes—
445C22×C4C22×D418yes—
446C22×C4C22×D418yes—
447C22×C4C22×D414yes—
448C22×C4C22×D414yes—
449C22×C4C22×D414yes—
450C22×C4C22×D412yes—
451C22×C4C22×D414yes—
452C22×C4C2×#4118yes—
453C22×C4C2×#4112yes—
454C22×C4C2×#4112yes—
455C22×C4C2×#4114yes—
456C22×C4C2×#4112yes—
457C22×C4C2×#4112yes—
458C22×C42−1+414yes—
459C4⋊C4—28no—
460C4⋊C4—232no—
461C4⋊C4—232no—
462C4⋊C4—232no—
463C4⋊C4—216no—
464C4⋊C4—216no—
465C4⋊C4—48no—
466C4⋊C4—416no—
467C4⋊C4—48no—
468C4⋊C4—416no—
469C4⋊C4—48no—
470C4⋊C4—88no—
471C22⋊C4—28no—
472C22⋊C4—28no—
473C22⋊C4—232no—
474C22⋊C4—232no—
475C22⋊C4—48no—
476C22⋊C4—416no—
477C22⋊C4—48no—
478C22⋊C4—48no—
479C22⋊C4—88no—
480C42—28no—
481C42—28no—
482C42—416no—
483C42—48no—
484C42—44no—
485C42—44no—
486C42—416no—
487C42#127116yes—
488C23—24no—
489C23—22no—
490C23—22no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—22no—
500C23—24no—
501C23—22no—
502C23—22no—
503C23—42no—
504C23—41no—
505C23—41no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—41no—
510C23—41no—
511C23—42no—
512C23—42no—
513C23—41no—
514C23—42no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23#54114yes—
530C23C23×D412yes—
531C23C23×D411yescharacteristic
532C23C22×#4111yescharacteristic
533C23C22×#4111yescharacteristic
534C23C22×#4112yes—
535C23C2×2−1+412yes—
536C23C2×2−1+412yes—
537C23#58512yes—
538C23#58512yes—
539Q8—432no—
540D4—416no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—28no—
551C2×C4—28no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—28no—
555C2×C4—28no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—28no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—28no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—28no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—432no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—44no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—44no—
584C2×C4—41no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—44no—
594C2×C4—41no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—416no—
604C2×C4—416no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4#52514yes—
620C2×C4#52514yes—
621C2×C4#52518yes—
622C2×C4#52518yes—
623C2×C4C2×#12718yes—
624C2×C4C2×#12714yes—
625C22—28no—
626C22—24no—
627C22—44no—
628C22—44no—
629C22—44no—
630C22—44no—
631C22—44no—
632C22—42no—
633C22—42no—
634C22—44no—
635C22—44no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22C2×#52512yes—
644C22C2×#52511yescharacteristic
645C22C2×#52512yes—
646C22C2×#52511yescharacteristic
647C22C22×#12711yescharacteristic
648C22C2×#54111yescharacteristic
649C22#345511yescharacteristic
650C22#346112yes—
651C22#348512yes—
652C22#348612yes—
653C22#348912yes—
654C22#348912yes—
655C22#348912yes—
656C22#348912yes—
657C22#349512yes—
658C22#350911yescharacteristic
659C22#351211yescharacteristic
660C22#351612yes—
661C22#351812yes—
662C22#352312yes—
663C22#352412yes—
664C4—28no—
665C4—44no—
666C4—48no—
667C4—44no—
668C4—48no—
669C4—42no—
670C4—42no—
671C4—42no—
672C4—42no—
673C4—48no—
674C4—82no—
675C4—84no—
676C4#239718yes—
677C2—44no—
678C2—82no—
679C2C2×#239711yescharacteristic
680C2#3515512yes—
681C2#3524412yes—
682C2#3530911yescharacteristic
683C2#3531011yescharacteristic
684C2#3563412yes—
685C2#3565412yes—
686C2#3566312yes—
687C2#3566512yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#2327C2semidirect
2. C2×#2323C2semidirect
3. C2×C4×C22⋊Q8C2semidirect
5. #15223C2semidirect
6. #15223C2semidirect
7. #15193C2semidirect
8. #15178C2semidirect
9. #15178C2semidirect
11. #15082C2semidirect
12. #15079C2semidirect
14. #14969C2semidirect
15. #13797C2semidirect
149. C4×C4⋊C4D4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#23272
C2×#23232
C2×C4×C22⋊Q82
C2×#22902
#152232
#151932
#151782
#151772
#150822
#150792
#149852
#149692
#137972